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Simplifying 2(x) = -0.5x2 + 1.10x + 4.2 Reorder the terms: 2x = 4.2 + 1.10x + -0.5x2 Solving 2x = 4.2 + 1.10x + -0.5x2 Solving for variable 'x'. Reorder the terms: -4.2 + 2x + -1.10x + 0.5x2 = 4.2 + 1.10x + -0.5x2 + -4.2 + -1.10x + 0.5x2 Combine like terms: 2x + -1.10x = 0.9x -4.2 + 0.9x + 0.5x2 = 4.2 + 1.10x + -0.5x2 + -4.2 + -1.10x + 0.5x2 Reorder the terms: -4.2 + 0.9x + 0.5x2 = 4.2 + -4.2 + 1.10x + -1.10x + -0.5x2 + 0.5x2 Combine like terms: 4.2 + -4.2 = 0.0 -4.2 + 0.9x + 0.5x2 = 0.0 + 1.10x + -1.10x + -0.5x2 + 0.5x2 -4.2 + 0.9x + 0.5x2 = 1.10x + -1.10x + -0.5x2 + 0.5x2 Combine like terms: 1.10x + -1.10x = 0.00 -4.2 + 0.9x + 0.5x2 = 0.00 + -0.5x2 + 0.5x2 -4.2 + 0.9x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 -4.2 + 0.9x + 0.5x2 = 0.0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -8.4 + 1.8x + x2 = 0 Move the constant term to the right: Add '8.4' to each side of the equation. -8.4 + 1.8x + 8.4 + x2 = 0 + 8.4 Reorder the terms: -8.4 + 8.4 + 1.8x + x2 = 0 + 8.4 Combine like terms: -8.4 + 8.4 = 0.0 0.0 + 1.8x + x2 = 0 + 8.4 1.8x + x2 = 0 + 8.4 Combine like terms: 0 + 8.4 = 8.4 1.8x + x2 = 8.4 The x term is 1.8x. Take half its coefficient (0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. 1.8x + 0.81 + x2 = 8.4 + 0.81 Reorder the terms: 0.81 + 1.8x + x2 = 8.4 + 0.81 Combine like terms: 8.4 + 0.81 = 9.21 0.81 + 1.8x + x2 = 9.21 Factor a perfect square on the left side: (x + 0.9)(x + 0.9) = 9.21 Calculate the square root of the right side: 3.034798181 Break this problem into two subproblems by setting (x + 0.9) equal to 3.034798181 and -3.034798181.Subproblem 1
x + 0.9 = 3.034798181 Simplifying x + 0.9 = 3.034798181 Reorder the terms: 0.9 + x = 3.034798181 Solving 0.9 + x = 3.034798181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + x = 3.034798181 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + x = 3.034798181 + -0.9 x = 3.034798181 + -0.9 Combine like terms: 3.034798181 + -0.9 = 2.134798181 x = 2.134798181 Simplifying x = 2.134798181Subproblem 2
x + 0.9 = -3.034798181 Simplifying x + 0.9 = -3.034798181 Reorder the terms: 0.9 + x = -3.034798181 Solving 0.9 + x = -3.034798181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + x = -3.034798181 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + x = -3.034798181 + -0.9 x = -3.034798181 + -0.9 Combine like terms: -3.034798181 + -0.9 = -3.934798181 x = -3.934798181 Simplifying x = -3.934798181Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.134798181, -3.934798181}
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